Integrand size = 79, antiderivative size = 30 \[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=-2-2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \]
\[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=\int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx \]
Integrate[(2^(-2 + x)^(-1)*(E^E^E^(3 + E^(6 + x)))^(-2 + x)^(-1)*(E^(9 + E ^(3 + E^(6 + x)) + E^(6 + x) + x)*(2 - x) + Log[2*E^E^E^(3 + E^(6 + x))])) /(4 - 4*x + x^2),x]
Integrate[(2^(-2 + x)^(-1)*(E^E^E^(3 + E^(6 + x)))^(-2 + x)^(-1)*(E^(9 + E ^(3 + E^(6 + x)) + E^(6 + x) + x)*(2 - x) + Log[2*E^E^E^(3 + E^(6 + x))])) /(4 - 4*x + x^2), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {2^{\frac {1}{x-2}} \left (e^{e^{e^{e^{x+6}+3}}}\right )^{\frac {1}{x-2}} \left (e^{x+e^{e^{x+6}+3}+e^{x+6}+9} (2-x)+\log \left (2 e^{e^{e^{e^{x+6}+3}}}\right )\right )}{x^2-4 x+4} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int \frac {2^{\frac {1}{x-2}-2} \left (e^{e^{e^{3+e^{x+6}}}}\right )^{\frac {1}{x-2}} \left (e^{x+e^{3+e^{x+6}}+e^{x+6}+9} (2-x)+\log \left (2 e^{e^{e^{3+e^{x+6}}}}\right )\right )}{(2-x)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 4 \int \left (\frac {2^{\frac {1}{x-2}-2} \left (e^{e^{e^{3+e^{x+6}}}}\right )^{\frac {1}{x-2}} \log \left (2 e^{e^{e^{3+e^{x+6}}}}\right )}{(x-2)^2}-\frac {2^{\frac {1}{x-2}-2} e^{x+e^{3+e^{x+6}}+e^{x+6}+9} \left (e^{e^{e^{3+e^{x+6}}}}\right )^{\frac {1}{x-2}}}{x-2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \left (-\int \frac {2^{\frac {1}{x-2}-2} e^{x+e^{3+e^{x+6}}+e^{x+6}+9} \left (e^{e^{e^{3+e^{x+6}}}}\right )^{\frac {1}{x-2}}}{x-2}dx-\int e^{x+e^{3+e^{x+6}}+e^{x+6}+9} \int \frac {2^{\frac {1}{x-2}-2} \left (e^{e^{e^{3+e^{x+6}}}}\right )^{\frac {1}{x-2}}}{(x-2)^2}dxdx+\log \left (2 e^{e^{e^{e^{x+6}+3}}}\right ) \int \frac {2^{\frac {1}{x-2}-2} \left (e^{e^{e^{3+e^{x+6}}}}\right )^{\frac {1}{x-2}}}{(x-2)^2}dx\right )\) |
Int[(2^(-2 + x)^(-1)*(E^E^E^(3 + E^(6 + x)))^(-2 + x)^(-1)*(E^(9 + E^(3 + E^(6 + x)) + E^(6 + x) + x)*(2 - x) + Log[2*E^E^E^(3 + E^(6 + x))]))/(4 - 4*x + x^2),x]
3.13.58.3.1 Defintions of rubi rules used
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
\[-2^{\frac {1}{-2+x}} \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{3+{\mathrm e}^{6+x}}}}\right )^{\frac {1}{-2+x}}\]
int((ln(2*exp(exp(exp(3)*exp(exp(6+x)))))+(2-x)*exp(3)*exp(6+x)*exp(exp(6+ x))*exp(exp(3)*exp(exp(6+x))))*exp(ln(2*exp(exp(exp(3)*exp(exp(6+x)))))/(- 2+x))/(x^2-4*x+4),x)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=-e^{\left (\frac {{\left (e^{\left (x + e^{\left (x + 6\right )} + 9\right )} \log \left (2\right ) + e^{\left (x + e^{\left (x + 6\right )} + e^{\left (e^{\left (x + 6\right )} + 3\right )} + 9\right )}\right )} e^{\left (-x - e^{\left (x + 6\right )} - 9\right )}}{x - 2}\right )} \]
integrate((log(2*exp(exp(exp(3)*exp(exp(6+x)))))+(2-x)*exp(3)*exp(6+x)*exp (exp(6+x))*exp(exp(3)*exp(exp(6+x))))*exp(log(2*exp(exp(exp(3)*exp(exp(6+x )))))/(-2+x))/(x^2-4*x+4),x, algorithm=\
-e^((e^(x + e^(x + 6) + 9)*log(2) + e^(x + e^(x + 6) + e^(e^(x + 6) + 3) + 9))*e^(-x - e^(x + 6) - 9)/(x - 2))
Timed out. \[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=\text {Timed out} \]
integrate((ln(2*exp(exp(exp(3)*exp(exp(6+x)))))+(2-x)*exp(3)*exp(6+x)*exp( exp(6+x))*exp(exp(3)*exp(exp(6+x))))*exp(ln(2*exp(exp(exp(3)*exp(exp(6+x)) )))/(-2+x))/(x**2-4*x+4),x)
Time = 0.45 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=-e^{\left (\frac {e^{\left (e^{\left (e^{\left (x + 6\right )} + 3\right )}\right )}}{x - 2} + \frac {\log \left (2\right )}{x - 2}\right )} \]
integrate((log(2*exp(exp(exp(3)*exp(exp(6+x)))))+(2-x)*exp(3)*exp(6+x)*exp (exp(6+x))*exp(exp(3)*exp(exp(6+x))))*exp(log(2*exp(exp(exp(3)*exp(exp(6+x )))))/(-2+x))/(x^2-4*x+4),x, algorithm=\
\[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=\int { -\frac {{\left ({\left (x - 2\right )} e^{\left (x + e^{\left (x + 6\right )} + e^{\left (e^{\left (x + 6\right )} + 3\right )} + 9\right )} - \log \left (2 \, e^{\left (e^{\left (e^{\left (e^{\left (x + 6\right )} + 3\right )}\right )}\right )}\right )\right )} \left (2 \, e^{\left (e^{\left (e^{\left (e^{\left (x + 6\right )} + 3\right )}\right )}\right )}\right )^{\left (\frac {1}{x - 2}\right )}}{x^{2} - 4 \, x + 4} \,d x } \]
integrate((log(2*exp(exp(exp(3)*exp(exp(6+x)))))+(2-x)*exp(3)*exp(6+x)*exp (exp(6+x))*exp(exp(3)*exp(exp(6+x))))*exp(log(2*exp(exp(exp(3)*exp(exp(6+x )))))/(-2+x))/(x^2-4*x+4),x, algorithm=\
integrate(-((x - 2)*e^(x + e^(x + 6) + e^(e^(x + 6) + 3) + 9) - log(2*e^(e ^(e^(e^(x + 6) + 3)))))*(2*e^(e^(e^(e^(x + 6) + 3))))^(1/(x - 2))/(x^2 - 4 *x + 4), x)
Time = 13.93 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {2^{\frac {1}{-2+x}} \left (e^{e^{e^{3+e^{6+x}}}}\right )^{\frac {1}{-2+x}} \left (e^{9+e^{3+e^{6+x}}+e^{6+x}+x} (2-x)+\log \left (2 e^{e^{e^{3+e^{6+x}}}}\right )\right )}{4-4 x+x^2} \, dx=-2^{\frac {1}{x-2}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^6\,{\mathrm {e}}^x}\,{\mathrm {e}}^3}}{x-2}} \]